The harmonic oscillator is described by the action functional $$J[x] = \int_{t_0}^{t_1}\left(mx'^2 −\frac{1}{2}kx^2\right) dt$$ where $m$ is the mass and $k$ is the spring constant.
a. Show that the equation of motion is $mx'' + \frac{1}{2}kx = 0$.
b. What is the Hamiltonian and what are Hamilton’s equations of motion?
c. What are the equilibria of the system?
I did problems a and b but I am stuck on c. I might be overthinking how to find the equilibria. Any help would be great!!!
On
To find the equilibrium point of the system you have to consider the potential energy associated to the Lagrangian of the functional $J[x]$ namely the Lagrangian is the difference between the kinetic energy and the potential energy : $L=T-V$ so in your problem, the potential energy is identified with the second term of your lagrangian, which is also physically correct since is the potential energy of an harmonic oscillator. So then you can apply Fermat's theorem to the potential energy V, by that I mean $V'(x)=0$, so you determine the critical point of the derivative. After that you must substitute the value you have found into the underived V(x) and see which is the sign of the $V(x)$, the sign of the $V(x)$ will then decide the nature of the critical point so the equilibrium of the system.
The system is in equilibrium when $x$ doesn't change, i.e. if $x''=0$. That yields a simple equation for $x$. The equilibrium is stable if $k\gt0$ and unstable if $k\lt0$.